Computational Geometry: Algorithms for Spatial and Geometric Problems

Fundamental Structures and Algorithms

The convex hull of a set of points is the smallest convex polygon (in 2D) or polyhedron (in 3D) that contains all the points. It can be visualized as the shape formed by stretching a rubber band around the points. Computing the convex hull is one of the most fundamental operations in computational geometry, with efficient algorithms like Graham scan (O(n log n) in 2D) and the quickhull algorithm (efficient in practice for both 2D and 3D). Convex hulls are used in collision detection, shape analysis, and as a preprocessing step for many other geometric algorithms.

Voronoi diagrams partition space into regions based on proximity to a set of sites. Each region contains all points closer to its site than to any other site. The dual of the Voronoi diagram is the Delaunay triangulation, which connects sites whose Voronoi regions share an edge. Delaunay triangulations have the property of maximizing the minimum angle among all possible triangulations of the same point set, making them the basis for high-quality mesh generation in finite element and finite volume methods.

Spatial data structures organize geometric objects for efficient queries. K-d trees partition space using axis-aligned hyperplanes, enabling nearest-neighbor queries in O(log n) average time. R-trees group nearby objects into bounding rectangles arranged in a balanced tree, supporting range queries and intersection tests for collections of geometric objects. Octrees (and their 2D counterpart, quadtrees) recursively subdivide space into eight (or four) equal regions, providing a hierarchical spatial decomposition used in adaptive mesh refinement, N-body simulations, and collision detection.

Mesh Generation

Mesh generation is one of the most important practical applications of computational geometry. Every finite element and finite volume simulation requires a mesh, a decomposition of the computational domain into simple elements, and the quality of this mesh directly affects the accuracy, convergence, and computational cost of the simulation.

Delaunay-based mesh generation starts with a Delaunay triangulation of boundary points and progressively refines it by inserting new points where elements are too large or too poorly shaped. Ruppert algorithm and its extensions produce triangular meshes with guaranteed minimum angles (typically at least 20 to 30 degrees), which is important because thin, elongated elements degrade the accuracy of finite element solutions. The provable quality guarantees of Delaunay refinement make it the preferred approach for many automatic mesh generation tools.

Advancing front methods build the mesh from the domain boundary inward. Starting from the boundary edges, new triangles (or tetrahedra in 3D) are formed one at a time, advancing the front of unmeshed region inward. These methods produce high-quality meshes near boundaries, which is important for resolving boundary layers in fluid dynamics, but can encounter difficulties when fronts from different boundaries collide in the interior.

Hexahedral meshing (generating meshes of brick-like elements in 3D) remains one of the unsolved grand challenges of computational geometry. While tetrahedral meshing is largely automated, hexahedral meshing for arbitrary geometries still requires significant manual effort. This matters because hexahedral elements often provide superior accuracy per degree of freedom in structural and fluid simulations.

Geometric Intersection and Boolean Operations

Computing intersections between geometric objects is fundamental to many scientific computing applications. Ray-surface intersection is the basis of ray tracing in computer graphics and radiation transport simulations. Line-segment intersection algorithms detect crossings in planar subdivisions and are used in geographic information systems. Surface-surface intersection is needed for constructive solid geometry (CSG), where complex shapes are built by combining simpler ones using Boolean operations (union, intersection, difference).

The sweep line algorithm, introduced by Bentley and Ottmann, efficiently finds all intersections among n line segments in O((n + k) log n) time, where k is the number of intersection points. This is much faster than the naive O(n squared) approach of testing all pairs when the number of intersections is small relative to the number of segments.

Applications in Scientific Computing

Finite element analysis depends on computational geometry for mesh generation, mesh quality assessment, and mesh adaptation. The geometric properties of mesh elements (aspect ratio, minimum angle, skewness) directly affect the accuracy and stability of the numerical solution. Adaptive mesh refinement uses geometric algorithms to identify regions needing finer resolution and to refine the mesh while maintaining element quality.

Molecular and materials science uses Voronoi tessellations to analyze atomic packing in crystals and glasses, compute volumes and surface areas of molecular cavities, and partition space for domain decomposition in parallel molecular dynamics. The Voronoi cell around each atom provides a natural definition of atomic volume that is used to compute local properties like stress, strain, and free volume.

Geographic information systems (GIS) rely on computational geometry for map overlay operations (intersecting layers of geographic data), point-in-polygon tests (determining which region contains a given location), buffer generation (creating regions within a specified distance of features), and triangulated irregular network (TIN) construction for terrain modeling.

Computer vision and 3D reconstruction use computational geometry to reconstruct surfaces from point clouds (sets of 3D coordinates measured by laser scanners or computed from stereo images). Alpha shapes generalize convex hulls to capture concavities and holes in the point cloud. Surface reconstruction algorithms like Poisson surface reconstruction convert noisy point clouds into smooth triangle meshes suitable for visualization and simulation.

Robotics and motion planning use computational geometry for path planning (finding collision-free paths through environments), workspace analysis (determining the set of reachable configurations), and sensor coverage optimization. Configuration space, a geometric abstraction that maps the robot degrees of freedom to a high-dimensional space where obstacles become forbidden regions, transforms motion planning into a geometric search problem.